Preheat oven to 350°F with rack in middle. Generously butter pan and dust with flour, knocking out excess.

Whisk together dry ingredients: flour, baking powder, baking soda, and salt. Beat together butter and sugar in an electric mixer at medium speed until pale and fluffy(about 5 minutes). Scrape seeds from vanilla beans with tip of a paring knife and add into butter mixture, reserving pods for another use, and beat until well combined (about 1 minute). Add eggs 1 at a time, beating well after each addition, then beat in lemon extract until well combined. At low speed ("stir" setting on KitchenAid mixer) add flour mixture and milk alternately in batches, beginning and ending with flour mixture, mixing until just combined.

Split vanilla bean and scrape seeds into whole milk. Let sit in a spouted container, such as a large pyrex measuring cup, for about an hour. Add confectioners' sugar gradually, whisking, until you get desired consistency--about 1 cup. You want to make sure that it's not too runny, or it won't dry white on the cake, and will run off too quickly. It should take a few seconds to whisk it, and it will feel too thick at first--keep whisking until you get desired thickness. If too thick, add a teaspoon or so of the vanilla milk.